\chapter*{Preface}

This is a course in mathematical proof. 
It is for math majors, typically sophomores in the US, although since
it requires only high school mathematics
it can be used with first year students.



\medskip
\noindent\textsc{Approach.}
This course is inquiry-based (sometimes called Moore method 
or discovery method).
This text is a sequence of exercises,
along with definitions and a few remarks.
Students work through the material together by
proving statements or by providing examples or counterexamples.
This makes each person grapple directly with the 
mathematics\Dash the instructor only 
lightly guides, while the students pledge not to use outside sources\Dash
talking out misunderstandings, 
sometimes stumbling in the dark, and sometimes
having beautiful flashes of insight.
For these students, with this material,
this is the best way to develop mathematical maturity.
Besides, it is a lot of fun.


\medskip
\noindent\textsc{Topics.}
We start with elementary number theory, not logic and sets, 
for the same reason
that the baseball team's annual practice starts with tossing the ball and 
not with reading the rulebook.
Math majors take readily to proving things about
divisibility and primes 
whereas a month of preliminary material is less of a lure.

But the background is good stuff also and 
students are on board once they see where it is going.
In the second and third chapters we do
sets, functions, and relations, keeping the
intellectual habits that we established at the start.



\medskip
\noindent\textsc{Exercises.}
As much as the material allows,
nearby exercises have about the same difficulty.
This standard gradually rises.

Some exercises have multiple items; these come in two types.
If the items are labeled \textsc{A}, \textsc{B}, etc., 
then each one is hard enough to be a separate assignment.
If the labels are (i), (ii), etc., then they together make
a single assignment.
I have students put proposed solutions on the board
for the group to discuss and
if the items are labelled alphabetically then I ask a different student
to do each one, while for the others I ask a single student to do them all.

This text comes in versions that differ in the number of exercises,
so it is adoptable for courses with different needs.
You are reading the \thisversion{} version.
For more, see the home page.
% 3*3*15=135
% 4*3*15=180


\medskip
\noindent\textsc{Home page.}
This book is Free; see \url{http://joshua.smcvt.edu/proofs}.
That site has other material related to this text, including 
its \LaTeX{} source.

\vspace*{.1in}
\vspace{\fill}
\noindent\parbox{.95\textwidth}{\raggedright\textit{At the first meeting of the class Moore would define the basic terms and either challenge the class to discover the relations among them, or, depending on the subject, the level, and the students, explicitly state a theorem, or two, or three. Class dismissed. Next meeting: "Mr Smith, please prove Theorem 1. Oh, you can't? Very well, Mr Jones, you? No? Mr Robinson? No? Well, let's skip Theorem 1 and come back to it later. How about Theorem 2, Mr Smith?" Someone almost always could do something. If not, class dismissed. It didn't take the class long to discover that Moore really meant it, and presently the students would be proving theorems and watching the proofs of others with the eyes of eagles.}\hspace{1.5em}---Paul Halmos}

\vspace{.1in}
\noindent\parbox{.95\textwidth}{\textit{The most important thing [is that] proving things in math [i]s a skill like any other that you get good at through practice.}\hspace{1.5em}---Cathy O'Neil}  % mathbabe blog

\vspace{.1in}
\noindent\parbox{.95\textwidth}{\textit{It's a kind of art that may change lives.}\hspace{1.5em}---Peter Schjeldahl}  % , \textit{New Yorker}
 
\vspace*{.15in}
\begin{flushright}
  \begin{tabular}{@{}l@{}}
  Jim Hef{}feron  \\
  Saint Michael's College  \\
  Colchester, Vermont USA \\
  2015-Spring
  \end{tabular}
\end{flushright}
